Analysis of cardiac rhythm using RR interval characterization

ABSTRACT

A method for analysis of cardiac rhythms, based on calculations of entropy and moments of interbeat intervals. An optimal determination of segments of data is provided that demonstrate statistical homogeneity, specifically with regard to moments and entropy. The invention also involves calculating moments and entropy on each segment with the goal of diagnosis of cardiac rhythm. More specifically, an absolute entropy measurement is calculated and provided as a continuous variable, providing dynamical information of fundamental importance in diagnosis and analysis. Through the present invention, standard histograms, thresholds, and categories can be avoided.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a division of application Ser. No. 12/594,842 filedOct. 6, 2009, now U.S. Pat. No. 8,983,584, which is a National Stage ofPCT Application No. PCT/US08/60021 filed Apr. 11, 2008 (published as WO2008/128034), which claims benefit under 35 U.S.C. § 119(e) to U.S.Provisional Patent Application Ser. No. 60/923,080 filed on Apr. 12,2007 and Ser. No. 60/998,664, filed on Oct. 12, 2007, which are herebyincorporated by reference in their entireties.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was not made in the course of federally sponsoredresearch or development.

THE NAMES OF THE PARTIES TO A JOINT RESEARCH AGREEMENT

This invention was not made in the course of joint research agreement.

BACKGROUND OF THE INVENTION

The present invention relates generally to the field of cardiology andin particular to detection and analysis of cardiac function. There is aserious need for detection of normal and abnormal cardiac rhythms usingheart rate (HR) or interbeat interval series.

Several common clinical scenarios call for identification of cardiacrhythm in ambulatory out-patients. For example, atrial fibrillation (AF)is a common arrhythmia that is often paroxysmal in nature. Decisionsabout its therapy are best informed by knowledge of the frequency,duration and severity of the arrhythmia. While implanted devices canrecord this information with great accuracy, non-invasive diagnosticdevices for recording electrocardiographic (EKG) signals are constrainedby the need for skin electrodes. Non-invasive devices for determiningheart rate from the pulse rate are not in common use because of reducedconfidence in detecting AF based on the heart rate series alone.Specifically, sinus rhythm with frequent ectopy is expected to sharemany time series features with AF, and thus be difficult to distinguish.In addition, many other transient cardiac arrhythmias cause short-livedsymptoms but are currently difficult to diagnose on the basis of heartrate time series alone.

Thus a need exists for confident diagnosis of normal and abnormalcardiac rhythms from heart rate time series such as would be provided bynon-invasive devices that do not use a conventional EKG signal. Since acommon and high-profile example of an abnormal cardiac rhythm is atrialfibrillation, its detection from heart rate time series alone is anobject of the present invention.

Atrial fibrillation is an increasingly common disorder of cardiac rhythmin which the atria depolarize at exceedingly fast rates. Even withnormal function of the atrioventricular (AV) node, which serves as thesole electrical connection between the atria and the ventricles andfilters the high frequency of atrial impulses, atrial fibrillation canresult in heart rates as high as 160 to 180 beats per minute. Whilethese fast rates, along with the lack of atrial contractile function,may or may not cause symptoms, atrial fibrillation carries with it therisk of stroke because the lack of concerted atrial contraction allowsblood clots to form. Thus the major emphases in treatment are conversionto normal sinus rhythm (NSR), control of heart rates, andanticoagulation to reduce the risk of stroke.

Patients with severe heart disease are at increased risk of ventriculartachycardia (VT) or fibrillation, and implantablecardioverter-defibrillator (ICD) devices are recommended to reduce theincidence of sudden cardiac death. ICDs are small battery-poweredelectrical impulse generators which are implanted in at-risk patientsand are programmed to detect cardiac arrhythmia and correct it bydelivering a jolt of electricity to the heart muscle. These patients arealso at high risk of atrial fibrillation leading to inappropriate ICDshocks. While dual chamber devices allow better AF detection because theatrial electrical activity is known, single lead ICDs must rely entirelyon the RR interval time series. There is a need to improve detection ofAF in short records to reduce inappropriate ICD shocks.

The current management paradigm for patients with atrial fibrillationemphasizes anticoagulation and both heart rate control and attempts toconvert to normal sinus rhythm (NSR). This is based on the findings ofrandomized clinical trials that showed no morbidity or mortalityadvantage to either rhythm control or rate control strategies as long asanticoagulation was maintained. Some principles that dominate thecurrent practice are: anticoagulation for life once even a singleparoxysm of AF has been detected in patients at risk for stroke; higherdoses of AV nodal-blocking drugs to lower average heart rates, and morefrequent AV junction ablation coupled with permanent electronic cardiacpacing; and cardioversion, anti-arrhythmic drugs and, if they fail, leftatrial catheter ablation procedures to restore and maintain sinusrhythm.

Decisions about these therapeutic options are best made if there isaccurate estimation of the proportion of time spent in AF, or the “AFburden.” Many patients with AF are elderly, and in some there is asubstantial risk of anticoagulation because of the propensity to fall.If indeed an episode of AF were truly never to recur, then the risk ofanticoagulation after a few months could legitimately be avoided. Thereis a need, therefore, for a continuous monitoring strategy to determinethe need for continued anticoagulation in patients thought to be free ofAF.

Many patients with AF are unaware of persistently fast ventricular ratesthat would lead the physician to alter medications or to consider AVjunction ablation therapy in conjunction with electronic ventricularpacing. This also demonstrates a need for a continuous monitoringstrategy that reports descriptive statistics of the heart rate duringAF.

Moreover, patients for whom rhythm control is attempted requirecontinuous monitoring to determine the success of the therapy, and theneed for further therapies if AF recurs.

Detection of AF can be accomplished with very high degrees of accuracy,if an intra-atrial cardiac electrogram from an implanted pacing lead ora conventional EKG signal from skin electrodes are available. Neither isas non-obtrusive as a device that records the time from one arterialpulse waveform to the next, but such a non-invasive device can provideonly the heart rate time series with no information about cardiacelectrical activity. Thus an algorithm and computer method for detectingAF in a heart rate or pulse rate series is a desirable goal.

Tateno and Glass developed a measure based on the reasoning thatdistinctive differences between AF and sinus (or other) rhythms lay inthe degree of overall variability. [See: K. Tateno and L. Glass,“Automatic detection of atrial fibrillation using the coefficient ofvariation and density histograms of RR and ΔRR intervals,” Med Biol EngComput vol. 39, 664-671, 2001.]

Tateno and Glass used the canonical MIT-BIH Holter databases(www.physionet.org) to develop empirical cumulative distributionfunctions (ECDFs) of heartbeat interval and heart rates, and to test thehypothesis that a new data set belongs to the AF group. The resultingalgorithm, which was based on 90% of the data sets, had diagnosticperformance in the remaining 10%, with Receiver Operating Characteristic(ROC) area of 0.98, and sensitivity and specificity over 95% at somecut-offs.

Generally, there are some potential barriers to widespreadimplementation of the Tateno and Glass approach:

First, the data were collected from a non-random sample of 23 AFpatients in the early 1980's when medical practices were different withregard to heart rate-controlling (HR-controlling) drugs andanti-congestive heart failure drugs. The average heart rate in theMIT-BIH Atrial Fibrillation Database is 96 beats per minute, compared to81 beats per minute in the contemporary University of Virginia HolterDatabase.

Second, much rests on histograms of intervals occurring within 30-minuteblocks that are arbitrarily segregated by the mean heart rate during the30 minutes. This approach is vulnerable to large changes in resultsbased on small differences in heart rates. Moreover, some ECDFsrepresent many more patients and data points than others. Choosinghistogram boundaries so that each represents the same proportion of theentire database has appeal, but still suffers from inescapable problemswhen such bright cut-offs are employed.

Third, episodes of AF lasting less than 30 minutes might be missedaltogether, if surrounded by very regular rhythms.

Fourth, the MIT-BIH arrhythmia database that was used for testing isrelatively small.

Fifth, in the Tateno and Glass approach there is no analysis of thedynamics (i.e., the order) of RR intervals. This is an especiallyimportant distinction between the Tateno and Glass approach and thepresent invention.

With respect to the detection of atrial fibrillation from interbeatintervals, the Tateno and Glass method employs a Kolmogorov-Smirnov (KS)test of ECDFs of observed sets of ΔRR intervals (the difference betweenone RR interval and the next) versus empirical histograms of ΔRRintervals during atrial fibrillation (AF) obtained from MIT-BIH AtrialFibrillation Database. The fundamental measurement is the largestdistance, also called the KS distance, between ECDFs of observed dataand a template data set. Large distances suggest that the data setsrepresent different cardiac rhythms. The KS distance method of Tatenoand Glass is designed to distinguish AF from normal sinus rhythm (NSR)and from other arrhythmias such as paced rhythm, ventricular bigeminyand trigeminy, and others. Formally, the parameter calculated is theprobability that the observed intervals arise from AF, thus smallp-values (PV) provide evidence that data is not AF. Tateno and Glasssuggest a cutoff of PV>0.01 as a diagnostic criterion for AF.

The 16 ECDFs of ΔRR intervals during AF are based on 10,062non-overlapping 50 point AF episodes segregated by ranges of the mean RRinterval distributed as shown in Table 1.

TABLE 1 Mean RR Segments 350-399 38 400-449 325 450-499 548 500-549 1179550-599 2114 600-649 1954 650-699 1256 700-749 913 750-799 386 800-849342 850-899 256 900-949 331 950-999 265 1000-1049 124 1050-1099 241100-1149 7

There are appealing features of this method. There is nonparametriccharacterization of ΔRR densities; the mean RR interval is incorporatedinto the analysis; and it distinguishes AF from normal sinus rhythm(NSR) and from other arrhythmias in the MIT-BIH arrhythmia database.

However, the current art presents further limitations, disadvantages,and problems, in addition to the general limitations noted above.

First, the mean RR interval is not included as continuous variable, butrather in ranges. It is an object of the present invention to addressthe need for a new method, which utilizes the mean RR interval as acontinuous variable.

Second, the empirical cumulative distribution function (ECDF) analysisis not dependent at all on non-AF rhythms. It is an object of thepresent invention to address the need for a new method wherein the ECDFanalysis is dependent on non-AF rhythms.

Third, the analysis requires a large amount of histogram data (>500,000data points) for implementation. It is an object of the presentinvention to address the need for a new method, which requiressignificantly less histogram data.

Fourth, the histograms for low (<400) and high (>1049) mean RR intervalsare based on very few segments. It is an object of the present inventionto address the need for a new method, which avoids this limitation.

Fifth, there are no histograms for extremely high (>1150) mean RRintervals. It is an object of the present invention to address the needfor a new method, which avoids this limitation.

Sixth, the data are not independent, invalidating the theoreticalp-value calculation. It is an object of the present invention to addressthe need for a new method, which utilizes independent data.

The long-felt need for a new method that addresses the limitations,disadvantages, and problems, discussed above, is evidenced by the manydatabases available for development and testing of new AF detectionalgorithms. Several databases have been used during the development andtesting of the present invention.

The MIT-BIH Atrial Fibrillation (AF) Database, which consists of 10-hourrecordings from 23 patients with AF. Each beat has been manuallyannotated as to its rhythm. In all, there are 299 segments of AF lastinga total of 91.59 hours (40%) and 510,293 beats. The database can bedivided into 21,734 non-overlapping 50-point records with followingdistribution: AF 8320, NSR 12171, other 735 and mixed 508. For modelingof binary outcomes, the database can be considered as 8824 50-pointrecords with any AF and 12,910 with no AF.

The MIT-BIH Arrhythmia (ARH) Database consists of two parts (100 seriesand 200 series) with 30-minute recordings (1500 to 3400 beats). The 100series contains 23 subjects (48244 total beats) with no AF, but someother abnormal rhythms (7394 beats). The 200 series contains 25 subjects(64394 total beats) with 8 subjects with AF (12402 beats, 11%); otherabnormal rhythms also present (13091 beats). The database can be dividedinto 2227 non-overlapping 50-point records with 289 (13%) having any AF.The overall distribution was AF 187, NSR 1351, other 255, and mixed 434.The development of new methods to detect atrial fibrillation has beenlimited, because the current go/no-go decision for developing new AFdetection algorithms rests on analysis of the ARH database, whichcontains only about 2 hours of AF in 8 patients from more than 20 yearsago.

Results obtained in the MIT-BIH databases may not hold up in widespreaduse because of their small sizes and highly selective nature.Accordingly, a more real-world data set of complete RR interval timeseries from consecutive 24-hour Holter monitor recordings has beenanalyzed.

The University of Virginia Holter Database consists of 426 consecutive24-hour recordings from the Heart Station beginning in October, 2005.206 are from males, and the median age is 58 years (10th percentile 23years, 90th percentile 80 years). 76 (18%) gave “atrial fibrillation”,“atrial fibrillation/flutter”, or “afib-palpitations” as the reason forthe test.

The dynamics of cardiac rhythms can be quantified by entropy and entropyrate under the framework of continuous random variables and stochasticprocesses [See C. E. Shannon, “A Mathematical Theory of Communication”,Bell System Technical Journal, vol. 27, pp. 379-423 & 623-656, July &October, 1948].

Approximate entropy (ApEn) was introduced in 1991 as a measure thatcould be applied to both correlated random and noisy deterministicprocesses with motivation drawn from the fields of nonlinear dynamicsand chaos theory [See: S. Pincus, “Approximate entropy as a measure ofsystem complexity,” Proc. Natl. Acad. Sci., vol. 88, pp. 2297-2301,1991.]. There are limitations and possible pitfalls in theimplementation and interpretation of ApEn, especially with the need todetect cardiac rhythms in relatively short data records.

Sample entropy (SampEn) is an alternative measure with betterstatistical properties and has successfully been utilized on neonatalheart rate data (HR data) to aid in the prediction of sepsis [See J.Richman and J. Moorman, “Physiological time series analysis usingapproximate entropy and sample entropy,” Amer J Physiol, vol. 278, pp.H2039-H2049, 2000; and D. Lake, J. Richman, M. Griffin, and J. Moorman,“Sample entropy analysis of neonatal heart rate variability,” Amer JPhysiol, vol. 283, pp. R789-R797, 2002.]. See also U.S. Pat. No.6,804,551 to Griffin et al. issued Oct. 12, 2004 and assigned to thesame assignee herein. The '551 patent is incorporated herein byreference in its entirety.

SampEn has also been used as part of the promising new multiscaleentropy (MSE) analysis technique to better discriminate adult HR dataamong normal, atrial fibrillation, and congestive heart failure patients[See: M. Costa, A. Goldberger, and C. Peng, “Multiscale entropy analysisof complex physiologic time series,” Phys. Rev. Lett., vol. 89, no. 6,p. 068102, 2002.]. For purposes of comparison, this work is termed thedeterministic approach to measuring complexity and order in heart ratevariability.

Standard error estimates aid in evaluating the adequacy of the selectedmatching tolerance r which can be especially crucial for short records.An expression for approximating the variance of sample entropy waspresented in D. Lake, J. Richman, M. Griffin, and J. Moorman, “Sampleentropy analysis of neonatal heart rate variability,” Amer J Physiol,vol. 283, pp. R789-R797, 2002, and used in selecting optimal values of mand r. Exploiting the special U-statistic structure of SampEn, thisestimate has recently been improved upon and asymptotic normalityestablished [See J. Richman, “Sample entropy statistics,” Ph.D.dissertation, University of Alabama Birmingham, 2004]. Estimating thestandard error for Apen and other Renyi entropy rate estimates hasproved to be more complicated because of the dependency of the randomprocess and the nonlinearities in the calculations.

With deterministic approaches the values of m and r are fixed for allthe analysis (sometimes signal length is also constant). This is done toenable comparison of a wider variety of processes, but has severaldisadvantages. The choices of m and r vary from study to study andcomparison of results is not always possible. Methods to optimallychoose these parameters have been studied and this process has been apart of developing entropy measures for detecting atrial fibrillation[See: D. Lake, J. Richman, M. Griffin, and J. Moorman, “Sample entropyanalysis of neonatal heart rate variability,” Amer J Physiol, vol. 283,pp. R789-R797, 2002.].

Current implanted devices employ a “stability” algorithm based on thevariability amongst a small number of interbeat or RR intervals, and“unstable” rhythms are interpreted as AF. The reasoning is that the mostdistinctive difference between AF and other rhythms lies in the degreeof variability.

BRIEF SUMMARY OF THE INVENTION

An aspect of various embodiments of the present invention system,computer method (and related computer, computer system, and/or computerprocessor) and computer program product provides for automatedclassification of cardiac rhythm, atrial fibrillation in particular,based only on the times between heart beats. In part, an algorithm orcomputer method, which is based on novel entropy measures, can be usedas a standalone diagnostic test for atrial fibrillation or as acomponent of multivariable diagnostic algorithms. The novel entropymeasures are called, coefficient of sample entropy (COSEn). According tothe present invention, the order of the heartbeat intervals as measuredby COSEn has diagnostic importance not provided in traditional tests foratrial fibrillation that are based on the heart rate and the degree ofvariability.

The diagnostic performance, which is robust for series as short as 12beats, is similar in the canonical MIT-BIH databases to the standardmethod developed by Tateno and Glass.

Various preferred embodiments of the present invention implement thecoefficient of sample entropy (COSEn) for detection of AF in shortrecords. COSEn is a measure of regularity that is optimized fordetection of AF in heart rate time series. COSEn incorporates sampleentropy, the conditional probability that two heart rate sequences oflength m, having matched within a specified tolerance, will also matchat the next point. The further modifications allow for different valuesof the tolerance allowed, and for the mean heart rate.

Implementation of COSEn as an AF detector has several advantages overexisting methods. For example, COSEn exploits information in theordering of heart beat intervals, a key difference between AF and otherrhythms. COSEn requires as few as 12 beats for accurate AF detection.COSEn is independent of heart rate—that is, works well at fast rates.COSEn adds statistically significant independent information tovariability measures.

As an example, FIG. 1 shows the 24-hour heartbeat time series recordedby a Holter monitor at the University of Virginia. Beats labeled asnormal sinus rhythm are in blue, and beats labeled as atrialfibrillation are in green. Note that the y-axis is RR interval, and AFis characterized by faster rates (shorter intervals) as well asincreased variability. The purple line at the bottom is the COSEn,calculated every 50 beats. The arrow marks a threshold above which thebeat would be labeled as atrial fibrillation. There is good agreementbetween this single statistical measure and the time series appearance,affirming the utility of the new measure as a diagnostic test for atrialfibrillation. There is good agreement between the data labels and thevalue of COSEn, with values above the threshold corresponding to epochsof AF.

The new algorithm and computer method, according to various embodimentsof the present invention, is designed for accurate detection of atrialfibrillation in implanted devices in which atrial activity is notmeasured, such as single lead implantable cardioverter-defibrillators,and in prolonged monitoring such as mobile cardiac outpatient telemetry.

A further aspect of various embodiments of the present inventionprovides a new computer methodology, system and computer program productof multivariate statistical models that employ entropy measures. Thisaspect answers some of the limitations and deficiencies discussed abovewith regard to previous methods. For example, in an embodiment of amethod, preferably a computer method, according to the presentinvention:

1) RR intervals are characterized by their dynamics, taking into accountthe order of the data points;

2) the novel dynamic parameters include differential quadratic Renyientropy rate measured using the SampEn algorithm, which is well-suitedfor missing points and small records;

3) both normalized sample entropy, denoted by SE, non-normalized sampleentropy, denoted by Q, and the coefficient of sample entropy (COSEn) aretaken into account; and

4) these new dynamical measures are combined with density parameterssuch as mean, standard deviation and coefficient of variation (CV)measured using histogram summary statistics—importantly, in the methodaccording to various embodiments of the present invention, theseparameters are considered as continuous values, not ranges.

In a further aspect of the present invention, method, system, andcomputer program product, atrial fibrillation is detected using amultivariable analysis such as a logistic regression model trained topredict the probability that a given segment of RR intervals is from apatient in atrial fibrillation. Other multivariable methods such asneural networks, nearest-neighbor analyses and others would also besuitable.

Results from the method of the present invention method have beencompared to the KS distance method of Tateno and Glass in the canonicaland publicly available MIT-BIH AF Database. Representative predictivemodels perform well in detecting AF, as assayed by receiver-operatingcharacteristics (ROC) areas as shown later. Note that the method ofvarious embodiments of the present invention only requires a fewcoefficients for implementation regardless of number of AF patients usedto train model. This is more efficient than repeated comparisons tomultiple histograms, as is called for in the KS distance method ofTateno and Glass.

Various embodiments of the present invention involve classifying CardiacRhythms using Moments and Entropy Rates of RR intervals.

The various algorithms used in embodiments of the present inventionovercome the potential problems of the prior art of classifying cardiacrhythm and detecting AF. Various aspects of the present invention may bebased on several fundamental differences between AF and other rhythms.The first is the variability itself—AF has more—and the second is theorder of the heartbeat intervals—AF has less. A third difference is thatthe blood pressure level and variability are altered. Thus, themeasurements used to classify cardiac rhythms fall into two basiccategories 1) estimates of the moments of RR intervals and transformedRR intervals 2) estimates of the entropy rate of the heart beat andtransformed heart beat series to characterize the heart rate dynamics.

The first category includes measurements that are associated withestablished statistical methods, such as the coefficient of variationand histograms of the RR intervals, but also includes the novel momentsafter the logarithmic or the unit mean transformation of heart rateinterbeat intervals. The second category includes the family of Renyientropy (or q-entropy) rates as described in Lake D E, Renyi entropymeasures of heart rate Gaussianity. IEEE Transactions on BiomedicalEngineering, Volume 53(1):21-27, 2006.

The variance and bias of entropy estimates becomes a significant issuefor short records, and an appropriate member of the family to emphasizeis differential quadratic entropy rates (q=2) which is denoted by Q andcalculated using the SampEn algorithm with optimized values of theparameters m, and r (See J. Richman and J. Moorman, “Physiological timeseries analysis using approximate entropy and sample entropy,” Amer JPhysiol, vol. 278, pp. H2039-H2049, 2000. and D. Lake, J. Richman, M.Griffin, and J. Moorman, “Sample entropy analysis of neonatal heart ratevariability,” Amer J Physiol, vol. 283, pp. R789-R797, 2002.).

One member of these families of measurements proves to be particularlystrong in discriminating atrial fibrillation from normal sinus rhythmand other arrhythmias and deserves being treated as a new thirdcategory, which we term coefficient of entropy (COE) measures. Thecoefficient of entropy is a calculation of an entropy rate (or entropy)of an RR interval series after it has been unit mean normalized(dividing each observation by the mean of the series). This is analogousto the coefficient of variation, which is the standard deviation afternormalization by the mean. In practice, the calculation of thecoefficient of entropy is accomplished by subtracting the naturallogarithm of the mean from the original entropy calculation. Thecoefficient of entropy calculated for Q in this way is especiallyeffective and we give it the name coefficient of sample entropy or COSEnfor short and denote it by Q*. A section describing the calculation ofthe new measure COSEn in more detail appears below.

An aspect of various embodiments of the present invention provides a newcomputer method, system and computer program product for analysis ofcardiac rhythm based on calculations of entropy and moments of interbeatintervals.

A first part of an embodiment of the present invention provides anoptimal determination of segments of data that are arbitrarily similarwith regard to moments and entropy. The second part of the new inventionconsists of further calculations of moments and entropy that areperformed on each segment with the goal of diagnosis of cardiac rhythm.

Some exemplary advantages associated with various embodiments of thepresent invention over current art include, but are not limited to, thefollowing: (1) determination of optimal segmentation of the data record,as opposed to fixed record lengths, (2) consideration of the order ofintervals, wherein important diagnostic information has been neglected,(3) simplicity of computation, (4) avoidance of comparisons tostandards, and (5) a single method producing a continuous measurement,rather than numerous methods relying on thresholds and categories.

These advantages provide solutions to problems inherent to the currentart of comparing histograms of interbeat interval differences tostandard histograms, which (1) are constrained by fixed record lengths,which may contain more than one rhythm, (2) do not utilize dynamicalinformation from the order of intervals, (3) require storage of standardhistogram data, (4) require consensus standards from large numbers ofwell-characterized patients, and (5) run the risk of large changes inresults based on very small differences in data that are near thresholdsof categories.

Embodiments of the present invention involve using entropy and momentcalculations to determine optimal segmentation of the data record forrhythm classification.

Embodiments of the present invention avoid using standard histograms,thresholds, and categories.

Embodiments of the present invention involve using an entropycalculation to make use of dynamical information of cardiac rhythm.Dynamical information of cardiac rhythm is of fundamental importance indiagnosis and analysis. Using dynamical information of cardiac rhythmprovides improved results over methods that rely only on distributionsof interbeat intervals without extracting information about the order inwhich they appear.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 shows a 24-hour Holter recording from a patient in NSR withparoxysmal AF, demonstrated by blue and green RR interval data points.

FIG. 2 shows frequency histograms of time series measures in 422 24-hourHolter monitor records from the University of Virginia Heart Station.

FIG. 3 shows a 24-hour Holter recording from a patient in NSR throughoutthe recording, demonstrated by blue RR interval data points. The purpleline at the bottom is COSEn, and the arrow marks the threshold abovewhich AF is diagnosed.

FIG. 4 shows a 24-hour Holter recording from a patient in AF throughoutthe recording, demonstrated by green RR interval data points. The purpleline at the bottom is COSEn, and the arrow marks the threshold abovewhich AF is diagnosed.

FIG. 5 shows AF detection performance in short RR interval time series.

FIG. 6 shows optimal segmentation of RR interval time series using anentropy-based method of the result for subject 202 in the MIT-BIHArrhythmia Database (See also FIG. 1 in K. Tateno and L. Glass,“Automatic detection of atrial fibrillation using the coefficient ofvariation and density histograms of RR and ΔRR intervals,” Med Biol EngComput vol. 39, 664-671, 2001.)

FIG. 7 shows mean of entropy rate estimation algorithms for 100simulations of Gaussian white noise (n=4096) with theoretical value of1.4189 for all m.

FIG. 8 shows the complete 30 minute RR interval time series of Record202 in Table 4.

FIG. 9 shows an explanation of Figure labels.

FIG. 10 shows EKG strips from labeled areas of FIG. 8.

FIG. 11 shows EKG strips from labeled areas of FIG. 8.

FIG. 12 shows a complete 30 minute RR interval time series from Record203 in Table 4.

FIG. 13 shows EKG strips from labeled areas of FIG. 12.

FIG. 14 shows histograms of COSEn of more than 700,000 16-beat segmentsfrom 114 24-hour records for which the rhythm labels of normal sinusrhythm (NSR) or AF were corrected.

FIG. 15 shows 24 hour RR interval data and rhythm analysis. AF is markedby open green bars (rhythm labeling from EKG inspection) and open purplebars (COSEn analysis).

FIG. 16 shows hour 2 of the 24 hour RR interval data and rhythm analysisshown in FIGS. 14 and 15.

FIG. 17 shows histograms of COSEn calculated in 16-beat segments fromthe entire MIT AF (top panels) and ARH (middle panels) databases andfrom the more than 100 UVa Holter recordings that we have over read(bottom panels). The left-hand panels are all rhythms other than AF, andthe right-hand panels are AF alone.

DETAILED DESCRIPTION OF THE INVENTION

Various embodiments of the present invention utilize entropy and entropyrate for analyzing rhythms, preferably cardiac rhythms.

The dynamics of cardiac rhythms can be quantified by entropy and entropyrate under the framework of continuous random variables and stochasticprocesses. The entropy of a continuous random variable X with density ƒisH(X)=E[−log(ƒ(X))]=∫_(−∞) ^(∞)−log(ƒ(x))ƒ(x)dxIf X has variance σ², then Y=X/σ has variance 1 and density σƒ(σy). Sothe entropy of Y is related to the entropy of X byH(Y)=∫_(−∞) ^(∞)−log(σƒ(σy))σƒ(σy)dy=H(X)=−log(σ)which shows that reduced entropy is indicative of reduced variance orincreased uncertainty.

Another important property of entropy is provided by the inequalityH(X)≤½(log(2πe)+log(σ²))=H(σZ)where Z is a standard Gaussian random variable. This result shows thatthe Gaussian distribution has maximum entropy among all random variableswith the same variance. Thus, an estimate of entropy that issubstantially lower than this upper bound for a random sample (withsample variance used as an estimate of σ²) provides evidence that theunderlying distribution is not Gaussian. This type of distribution is acharacteristic of some cardiac arrhythmias, such as bigeminy andtrigeminy, that are multimodal and is another reason entropy isimportant for this application.

Letting X denote the random sequence X₁, X₂, X₃, . . . , the entropyrate of X is defined as

${H(X)} = {\lim\limits_{n\rightarrow\infty}\frac{H\left( {X_{1},X_{2},\ldots\mspace{14mu},X_{n}} \right)}{n}}$where the joint entropy of in random variables X₁, X₂, . . . , X_(m) isdefined asH(X ₁ ,X ₂ , . . . ,X _(m))=E[−log(ƒ(X ₁ ,X ₂ , . . . ,X _(m)))]and ƒ is the joint probability density function ƒ. For stationaryprocesses, an equivalent definition is

${H(X)} = {{\lim\limits_{m->\infty}{H_{m}(X)}} = {\lim\limits_{m->\infty}{H\left( {{X_{m + 1}❘X_{1}},X_{2},\ldots\mspace{14mu},X_{m}} \right)}}}$so entropy rate is the entropy of the conditional distribution of thepresent observation given the past. The entropy rate for i.i.d.sequences reduces to the entropy of the common distribution.

Estimating the entropy rate for sequences depends on estimates of itsdensities of order m. Let X₁, X₂, . . . , X_(n) denote a stationaryrandom sequence and X_(i)(m) denote the template consisting of the m×1vector (X_(i−m+1), X_(i−m), . . . , X_(i))^(T). For notationalsimplicity, let X_(n)=X_(n)(n) denote the whole sequence and X=X_(∞)denote the limiting infinite sequence. The sequence X_(m)(m),X_(m+1)(m), . . . , X_(n)(m) is not independent, but many methodsdeveloped to analyze independent vector data are applicable. Inparticular, the m^(th)-order probability density function of thesequence, ƒ, and entropyE[−log(ƒ(X₁,X₂, . . . ,X_(m)))]can still be estimated empirically. These are the fundamentalcalculations in ApEn and SampEn.

The log-likelihood of a random sequence X_(n) can be written as

$\begin{matrix}{{\log\;{L\left( X_{n} \right)}} = {\log\left( {f\left( {X_{1},X_{2},\ldots\mspace{14mu},X_{n}} \right)} \right)}} \\{= {\sum\limits_{i = 1}^{n}{\log\left( {f\left( {{X_{i}❘X_{1}},X_{2},\ldots\mspace{14mu},X_{i - 1}} \right)} \right)}}}\end{matrix}$and the Shannon-McMillan-Breiman theorem [13] states that for stationaryergodic processes the entropy rate of X is related to the log-likelihoodfunction by

${H(X)} = {\lim\limits_{n->\infty}{\frac{1}{n}{E\left\lbrack {{- \log}\;{L\left( X_{n} \right)}} \right\rbrack}}}$where the convergence is with probability 1. As part of our invention,we use the term model-based entropy to indicate the estimate

$\hat{H} = {{H\left( {X_{n},\hat{\theta}} \right)} = {{- \frac{1}{n}}\log\;{L\left( {X_{n};\hat{\theta}} \right)}}}$obtained by modeling X by a parameter θ estimated by the MLE {circumflexover (θ)}. In the current application, X represents a sequence from aparticular cardiac arrhythmia that follows a particular parametricmodel.

All traditional time-series models, such as autoregressive (AR) models,could be applied to cardiac arrhythmias under this approach. In additionto increased flexibility, there is an important connection between thisand the current art. In particular, ApEn corresponds to model-basedentropy where the parameters are the transition probabilities of aMarkov chain of order m and they are estimated empirically.

According to preferred embodiments of the present invention, thedetection of cardiac rhythms is based on a series of the interbeat or RRintervals, which arise from a complex combination of both deterministicand stochastic physiological processes.

A complementary approach included as part of this invention is toconsider HR data sufficiently stochastic to model it as a randomprocess. We have developed stochastic Renyi entropy rate measures thatcan be reliably estimated with known statistical properties.

Embodiments of the present invention involve consideration of thestandard error of Renyi Entropy Rate Estimates.

It is important to know the standard error of entropy rate estimates inorder to be able to assess significant differences between cardiacrhythms and conduct meaningful statistical inference.

To demonstrate the novel approach of this invention, consider estimatingthe entropy rate. Letting Ĥ_(i)=−log({circumflex over (ƒ)}_(i)), theentropy rate estimate Ĥ_(i) is the mean of the sample Ĥ₁, Ĥ₂, . . . ^ .. . which can be viewed as an observation from a stationary randomprocess. Let {circumflex over (σ)}² denote the sample variance and ĉ_(k)denote the sample correlation coefficient at lag k calculated using adivisor of n_(k)=n−k, the number of pairs of observations at lag k. Thenthe variance of the entropy estimate can be estimated by

$\sigma_{\hat{H}}^{2} = {\frac{{\hat{\sigma}}^{2}}{n}\left( {1 + {2{\sum\limits_{k = 1}^{K}{n_{k}{\hat{c}}_{k}}}}} \right)}$and K is the maximum lag at which the random process has significantcorrelation. The optimal determination of K is application dependent,but our invention currently uses a conservative approach of selectingthe value that results in the largest variance.

This same general approach can be used to estimate the standard errorsof conditional Renyi entropy rates. In this case, the result comes fromanalyzing the sequence {circumflex over (ƒ)}₁ ^(q−1), {circumflex over(ƒ)}₂ ^(q−1), . . . ^ . . . . An estimate {circumflex over (σ)}_(q) ² ofthe variance can be calculated using the same expression as above withthe sample variance and correlation coefficients calculated from thissequence. Then, the standard error of the entropy estimated isapproximated by

$\sigma_{{\hat{R}}_{q}^{*}} = \frac{{\hat{\sigma}}_{q}}{{{q - 1}}{\hat{\mu}}_{q}}$where {circumflex over (μ)}_(q) is the sample mean of the sequence.

The stochastic Renyi entropy rate measures according to the presentinvention can be interpreted in ways that are analogous to thedeterministic concepts of complexity and order, and is not fundamentallydifferent. While developed under a stochastic framework, the algorithmsare easily modified to compute deterministic approach measures thatinclude both ApEn and SampEn. There are several basic differencesbetween the stochastic approach and the deterministic approaches, andeach has potential application to the detection of cardiac rhythms.

First, the deterministic approach involves calculating probabilitieswhile the stochastic approach calculates probability densities. Theprobabilities involve matches of templates of length m within atolerance r and converting them to densities by dividing by the volumeof the matching region, which is (2r)^(m). This simply reduces to addinga factor of log(2r) to ApEn or SampEn. The stochastic approach becomesviable when the values converge as r tends to 0 and the deterministicapproach is diverging.

Various embodiments of the present invention use both fixed value ofr=50 msec as well as r=f(S.D.). With fixed values, there is always thepossibility of encountering data that results in highly inaccurateentropy estimates, so included in our invention is the continueddevelopment of absolute entropy measures independent of m and r that arestatistically reliable and allows for comparison between a wide range ofHR data sets.

With the stochastic approach, the goal is to estimate the theoreticallimiting value as r goes to zero. The value of r for estimation does notneed to be fixed and can be optimized for each signal. In addition, forlonger records embodiments of the present invention also include in theinvention the option of not fixing m and instead estimating thetheoretical limiting value as m tends to infinity. One advantage of thisgeneral philosophy is that tolerances and template lengths can beselected individually for each signal to ensure accurate estimates. Evenif it is advantageous or necessary to compare signals at the same valueof r, our invention allows the flexibility of using different tolerancesfor estimation and applying a correction factor.

This idea is particularly important in the current setting of estimatingentropies of quantized RR intervals obtained from coarsely sampled EKGwaveforms. Another issue commonly encountered in analyzing RR intervaldata is that of quantization at the resolution of the sampling rate ofthe EKG signal. This means that all tolerances r within the resolutionwill result in the exact same matches and the issue becomes what value rshould be used to calculate the entropy rate. The proper choice is topick the value midway between the quantized values of r. For example,the EKG signal was sampled at 250 HZ in the AF Database and the RRintervals are at a resolution of 4 milliseconds. In this case, alltolerances between, say, 12 and 16 milliseconds would be considered 14for the log(2r) term. This continuity correction can be nontrivial whentolerances are close to the resolution of the data. This is a novelaspect of our invention that optimizes the accuracy and discriminatingcapability of the entropy measures.

Various embodiments of the present invention relate to calculatingCoefficient of Sample Entropy (COSEn), and preferably to AF detectionusing COSEn, in short records.

For patients with severe heart disease, increased risk of ventriculartachycardia (VT), or fibrillation, and especially for patients withimplantable cardioverter-defibrillator (ICD) devices, rapidity ofdiagnosis is paramount. Thus, embodiments of the present inventionquantify the diagnostic performance of COSEn over short record lengthsin comparison to a common variability measure, the coefficient ofvariation CV. FIG. 5 is a plot of ROC area as a function of recordlength for AF detection performance comparing COSEn (Q*) to CV on the AFand ARH databases for all possible overlapping records. In FIG. 5, theordinate is receiver-operating characteristic (ROC) area. In the inset,CV is coefficient of variation and Q* is COSEn. AF is the MIT-BIH AtrialFibrillation Database, and ARH is the MIT-BIH Arrhythmia Database. UsingCV, the ROC areas for detecting AF are 0.8 to 0.9, and change little forsequences between 4 and 25 beats in length. The ROC areas using COSEnare higher, especially when 10 or more beats are considered.

The improved performance of COSEn was evident even for records withfewer than 10 beats, and remained significantly higher than theperformance of CV for lengths as short as 5 beats.

We define the COSEn as the sample entropy of a series after beingnormalized by the mean. This is equivalent to subtracting the naturallogarithm of the mean from the original entropy. To see this, note thatif X has mean μ, then Y=X/μ has mean 1 and density μƒ(μy). So theentropy of Y is related to the entropy of X byH(Y)=∫_(−∞) ^(∞)−log(μƒ(μy))μƒ(μy)dy=H(X)−log(μ)as stated. Similar results can be shown for all Renyi entropy rates andin particular for the differential quadratic entropy rate Q calculatedusing the SampEn algorithm. This leads the calculationQ*=Q−log(μ)where Q* is the coefficient of sample entropy. As shown below, thismodification of entropy rate provides a very powerful univariatestatistic for classifying AF as part of this invention. In practice, themean can be estimated with the sample mean or sample median or otherrobust measures that minimize the effect of noisy, missed, and addedbeats.

Embodiments of the present invention involve Estimating Entropy UsingMatches.

To effectively detect cardiac rhythms, there is a need to be able toprocess short records of RR intervals that possibly includes missedbeats due to noise in the EKG or other limitations of the heartmonitoring device. An aspect of various embodiments of the presentinvention includes novel methods to accurately estimate entropy in thissetting using the intuitive notion of matches. A match occurs when allthe components of between two distinct templates X_(i)(m) and X_(j)(m)are within a specified tolerance r. The total number of matches fortemplate X_(i)(m) is denoted by A_(i)(m). For m=0, A_(i)(0) is equal tothe maximum number of possible matches which is n if self-matches areincluded and otherwise n−1. An estimate of the conditional probabilityof X_(i) given X_(i−1)(m) is

${\hat{p}}_{i} = {{{\hat{p}}_{i}(m)} = \frac{A_{i}\left( {m + 1} \right)}{A_{i - 1}(m)}}$and the corresponding estimate of the density is{circumflex over (ƒ)}_(i) ={circumflex over (p)} _(i)/(2r)The estimate of the entropy rate becomes

$\hat{H} = {{{- \frac{1}{n}}{\sum\limits_{i = 1}^{n}{\log\left( {\hat{f}}_{i} \right)}}} = {{{- \frac{1}{n}}{\sum\limits_{i = 1}^{n}{\log\left( {\hat{p}}_{i} \right)}}} + {\log\left( {2r} \right)}}}$and${\hat{R}}_{q} = {{\frac{1}{1 - q}{\log\left( {\frac{1}{n}{\sum\limits_{i = 1}^{n}{\hat{p}}_{i}^{q - 1}}} \right)}} + {\log\left( {2r} \right)}}$is the estimate of the general conditional Renyi entropy rate. In thesums above, all observations are shown while the conditional probabilityestimates are not always defined. In this case, they can be defined bysome convention or left out of the sum with the option of adjusting thedivisor n to reflect these omissions.

The analog of sample entropy, i.e. the quadratic differential entropyrate, is estimated by

$\hat{Q} = {{- {\log\left( \frac{A(m)}{B(m)} \right)}} + {\log\left( {2r} \right)}}$where ${A(m)} = {\sum\limits_{i = 1}^{n}{A_{i}\left( {m + 1} \right)}}$and ${B(m)} = {\sum\limits_{i = 1}^{n - 1}{A_{i}(m)}}$are the total number of matches of length m+1 and m. Note that most allof the above expressions involve slightly modified manipulation of thefundamental summary statistics A_(i)(m).

These estimates involve taking logarithms of ratios that becomeinaccurate or undefined when the numerator and or the denominator arenot sufficiently large. This becomes less likely an issue using thetotal number of matches, and this is a main reason that SampEn hasproven to be a more reliable and robust statistic for analyzing heartrate variability. Self-matches are included in the definition of ApEn toovercome problems of infinite or indeterminate ratios, but it still cansuffer from significant loss of accuracy when the number of matches issmall.

In order to improve the accuracy of ApEn and other conditional entropyrates, we introduce an algorithm that only calculates ratios withspecified minimum values of the numerator and denominator, denotedrespectively by n₀ and d₀. The conditional distribution of eachobservation can be calculated using increasing number of previousobservations, but at some template length the number of matches fallbelow the prescribed minimum. To avoid this possibility, we define a newconditional density estimate

f̂_(i)^(*)(m) = f_(i)(m^(*)) where$m^{*} = {{m^{*}(i)} = {\max\limits_{0 \leq k \leq m}\left\{ {{{A_{i}\left( {k + 1} \right)} \geq n_{0}},{{A_{i - 1}(k)} \geq d_{0}}} \right\}}}$for each m and observation i. This algorithm ensures that eachindividual contribution to the sums in (30) or (32) has some minimumdegree of statistical reliability. This also enables the inclusion oflong template matches when they are present and facilitates the goal ofnot fixing m and estimating the limiting parameters.

Embodiments of the present invention involve estimating the StandardError of Sample Entropy and COSEn.

The above methods, which apply to calculations of conditional entropyrates, need to be expanded slightly to allow for application to methodsusing the sample entropy. Recall that for a sequence of data, a set of mconsecutive points is called a template and can be considered a vector.An instance where all the corresponding components of two such vectorsare within a distance r of each other is called a match. Let B_(i) andA_(i) denote the number of matches with templates starting with thei^(th) point of the sequence of lengths m and m+1. respectively. Thenthe total number of matches of length m and m+1 are

$A = {\sum\limits_{i = 1}^{N - m}A_{i}}$ and$B = {\sum\limits_{i = 1}^{N - m}{B_{i}.}}$The conditional probability p of a match of length m+1. given a match oflength m can then be estimated by p=A/B. As presented in [16], thestandard error of p can be estimated by

$\sigma_{p}^{2} = {\frac{\sigma_{A}^{2}}{B^{2}} - {2A\frac{\sigma_{AB}^{2}}{B^{3}}} + {A^{2}\frac{\sigma_{B}^{2}}{B^{4}}}}$where σ_(A) ² is the variance of A, σ_(B) ² is the variance of B, andσ_(AB) ² is the covariance between A and B. The sample entropy is equalto −log(p) and the corresponding estimate of the standard error isσ_(p)/p.

Using a methodology similar to that introduced in Lake D E, Renyientropy measures of heart rate Gaussianity. IEEE Transactions onBiomedical Engineering, Volume 53(1):21-27, 2006, results in thefollowing estimates

$\sigma_{A}^{2} = {\frac{N - m}{4}{\sum\limits_{k = {- K}}^{K}{\sum\limits_{i = 1}^{N - k}{\left( {A_{i} - \overset{\_}{A}} \right)\left( {A_{i + k} - \overset{\_}{A}} \right)}}}}$$\sigma_{AB}^{2} = {\frac{N - m}{4}{\sum\limits_{k = {- K}}^{K}{\sum\limits_{i = 1}^{N - k}{\left( {A_{i} - \overset{\_}{A}} \right)\left( {B_{i + k} - \overset{\_}{B}} \right)}}}}$$\sigma_{B}^{2} = {\frac{N - m}{4}{\sum\limits_{k = {- K}}^{K}{\sum\limits_{i = 1}^{N - k}{\left( {B_{i} - \overset{\_}{B}} \right)\left( {B_{i + k} - \overset{\_}{B}} \right)}}}}$where Ā=A/(N−m), B=B/(N−m), and K is the maximum lag at which thesequences {A_(i)} and {B_(i)} have significant correlation. Theseestimates differ from those provided in [16] in that they do not fullyaccount for all the dependencies present in the calculations. Theadvantage of these expressions, especially for processing large amountsof data as is done in this paper, is that they require less computationand preliminary comparison of the methods suggest that they agreefavorably with the more accurate method.

The optimal determination of K is application dependent, but the natureof the calculations suggests that a minimum value of m is required. Aconservative approach used in this work selects the value that resultsin the largest variance. The factor of 4 in the above expressions comesfrom the fact that the expressions for A and B count each match twice.

The coefficient of sample entropy involves a first term with the sampleentropy and second term involving the natural logarithm of the samplemean. The standard deviation of COSEn to take into account theadditional variation of this second term which can be reasonably assumedto be uncorrelated with the sample entropy term. The standard deviationof the sample mean x can be estimated by s/√{square root over (n)} wheres is the sample standard deviation and n is the length of the segment ofRR intervals being analyzed. Then the standard deviation of Q*, denotedby σ*, can be estimated by:

$\sigma^{*} = {\sqrt{\frac{\sigma_{p}^{2}}{p} + \frac{s^{2}}{n{\overset{\_}{x}}^{2}}} = \sqrt{\frac{\sigma_{p}^{2}}{p} + \frac{{CV}^{2}}{n}}}$where CV is the coefficient of variation.

To demonstrate the robustness of the new algorithm, FIG. 7 shows themean results of estimating the entropy rate for 100 simulations ofGaussian white noise (n=4096). The tolerance r was set to 0.2 times thesample standard deviation for these and other entropy estimates shownbelow. The variants of the algorithms shown are no restrictions on thedenominator (d₀=1) with self-matches, d₀=10 with self-matches, and d₀=10without self-matches. In all cases, n₀=1 to avoid taking the logarithmof 0. The standard error for all these estimates is less than 0.002. Theunrestricted estimate is analogous to the traditional ApEn algorithmwhich clearly starts to rapidly degenerate after m=2. With d₀=10 and thenew algorithm, the estimates converge to stable values of on average1.143 with self-matches and 1.433 without. This latter result agreesfavorably with the theoretical value of (log(2π)+1)/2=1.419 and is anindication of the improved accuracy of the new invention.

The present invention involves segmentation of heart rate data intohomogeneous cardiac rhythms.

Characterizing heart rate data containing 2 or more different rhythmspresents a significant challenge. While analyzing short records helps tomitigate this problem, a better solution is to restrict analysis tosegments that have been identified as likely containing a homogeneouscardiac rhythm. Homogeneity can be defined, for example, in terms of themean, standard deviation, or other parameters of the RR intervaldistribution. Additionally, characteristics of the dynamics of asegment, such as the entropy or correlation function, can be parametersto consider. For a set of parameters, the homogeneity of a segment canbe measured based on the “goodness of fit” with some objective functionwhich increases as a segment becomes more nonhomogeneous. A simpleexample would be the sum of squared deviations of the RR intervals fromthe mean of a segment. Note that the self-matches, d₀=1 curve,corresponding to the traditional approximate entropy algorithm, quicklydegenerates after m=2. In the same spirit as impurity measures forClassification and Regression Trees (CART) and wavelet packets, thisobjective function can be termed “entropy.” To accomplish this goal, weemploy a novel method that automatically optimally divides heart ratedata into homogeneous segments. The methodology will be based on analgorithm we call Minimum Piecewise Entropy. Minimum Piecewise Entropywas originally developed to detect transient sonar signals and isdescribed below.

The approach is to hypothesize that the data is made up of some numberof homogeneous segments and to optimally estimate the number k andlocation of change points where the process is altered in some mannersuch as a shift in the mean. If the data is already homogeneous, thealgorithm should ideally estimate k to be 0 and no segmentation wouldoccur. The entropy of each stationary segment of data is calculated andthe criteria for optimality will be the piecewise entropy of the datawhich is calculated as the sum of the individual components. Asmentioned above, entropy could be any of a variety of measures includingthe sum of the squares of the residuals or the log-likelihood functionafter a particular model has been fit to the data.

Once a criterion has been determined, the problem becomes how to selectamong the all the possible ways a set of N points could be partitionedinto k segments.

Fortunately, there exists an efficient dynamic programming algorithm todo this optimization. To see this let E(i,j) denote the minimum entropyfor x(i), x(i+1), . . . , x(j). Also define e(j)=E(1,j)=minimumpiecewise entropy for x(1), x(2), . . . , x(j). The minimum piecewiseentropy can be found efficiently using dynamic programming because theentropy is assumed additive, that is, E(i,j)=E(i,k)+E(k+1,j). Therecursive algorithm to find the minimum piecewise entropy e is e(0)=0ande(j)=min_(1≤i≤k) {e(i−1)+E(i,j)}for j=1, 2, . . . , N.

All else being equal, fewer change points are preferable in estimatingthe piecewise entropy. The algorithm for minimum piecewise entropy canbe modified slightly to estimate the entropy using a certain number ofchange points. Define e(j,k)=minimum piecewise entropy for x(1), x(2), .. . , x(j) using k change points and the recursive algorithm generalizesto e(0,k)=0 ande(j,k)=min_(1≤i≤k) {e(i−1,k−1)+E(i,j)}for k=0, 1, . . . ,K and j=1, 2, . . . , N where K is some specifiedupper bound.

EXAMPLES

Various embodiments of the present invention, and the improved resultsobtained therefrom, are illustrated by way of the following,non-limiting examples.

Results from the method of the present invention method have beencompared to the KS distance method of Tateno and Glass in the canonicaland publicly available MIT-BIH AF Database. Representative predictivemodels perform well in detecting AF, as assayed by receiver-operatingcharacteristics (ROC) areas as shown later. Note that the method ofvarious embodiments of the present invention only requires a fewcoefficients for implementation regardless of number of AF patients usedto train model. This is more efficient than repeated comparisons tomultiple histograms, as is called for in the KS distance method ofTateno and Glass.

Example 1 Relates to AF Detection in the MIT-BIH Databases

In these examples univariable and multivariable methods were used toclassify cardiac rhythms, employing logistic regression and itsvariations. Segments of 50-point non-overlapping records are separatedinto binary outcomes with 1 denoting normal sinus rhythm or 0 denoting acardiac arrhythmia such as atrial fibrillation. A variety of the momentand entropy rate parameters described above are estimated for eachrecord and cardiac rhythm classifiers are developed using an optimalparsimonious subset of variables.

For purposes of this example, an optimal subset of variables for theMIT-BIH Atrial Fibrillation Database is the quadratic differentialentropy rate (Q), the mean (μ), and the standard deviation (σ) of the RRintervals. The entropy rate is calculated using the SampEn algorithmwith parameters m=1 and r=50 milliseconds. This result aided in thedevelopment of the coefficient of sample entropy (Q*) which is describedin more detail below. We also compare these results with the coefficientof variation CV=σ/μ

Subsets of parameters are evaluated using the significances ofindividual coefficients and of the overall model using the Waldstatistic adjusted for repeated measures. The overall significance ofthe model can be converted to a Wald Z-statistic which can be used tomake a fair comparison among models with varying number of parameters.The models are also verified on the independent MIT-BIH ArrhythmiaDatabase. The database is divided into 2075 non-overlapping 50 pointrecords with 184 (8.9%) AF records.

The results of this analysis are summarized below in Table 2. Theparameter TG represents results using the KS distance method of Tatenoand Glass.

TABLE 2 Model Performances on MIT-BIH AF data base Parameters AF ROCWald Wald Z ARH ROC log(CV) 0.913 25.0 17.0 0.862 Q 0.988 85.2 59.60.976 TG 0.992 38.1 26.3 0.976 Q, log(μ), log(σ) 0.995 82.4 32.4 0.985Q* 0.995 59.1 41.1 0.985

Example 2—AF Detection in 422 Consecutive Holter Monitor Recordings

In 920,242 50-beat records we calculated CV, KS distance (that is, weimplemented the method of Tateno and Glass), and COSEn. FIG. 2 showsfrequency histograms of the results, i.e., of time series measures in422 24-hour Holter monitor records from the University of Virginia HeartStation.

The multimodal nature suggests that different components contribute tothe overall distribution. We dissected the components using sums of 3Gaussians functions, shown as smooth lines representing an expression ofthe form:

${{f(x)} = {{\frac{A_{1}}{\sigma_{1}\sqrt{2\pi}}e^{{{- {({x - \mu_{1}})}^{2}}/2}\sigma_{1}^{2}}} + {\frac{A_{2}}{\sigma_{2}\sqrt{2\pi}}e^{{{- {({x - \mu_{2}})}^{2}}/2}\sigma_{2}^{2}}} + {\frac{A_{2}}{\sigma_{2}\sqrt{2\pi}}e^{{{- {({x - \mu_{2}})}^{2}}/2}\sigma_{2}^{2}}}}},$where A is the proportion of the total area attributed to eachcomponent, and μ and σ are the mean and standard deviation of eachcomponent.

The largest component is attributed to normal sinus rhythm (NSR), andthe next largest is attributed to atrial fibrillation (AF). Thesmallest, which always is intermediate in location, is attributed topremature ventricular contractions (PVCs) and premature atrialcontractions (PACs). These assignments are borne out qualitatively byinspection of individual records. A limitation, though, to this analysisis that we have not verified the rhythm labels of every beat. We knowfrom inspection of some of the records, that the labeling is notaltogether accurate. In each case, the numerical analyses were correctin classifying the rhythm label.

There was reasonable agreement about the relative proportions of rhythmlabels using the CV, KS distances, and COSEn. The proportions of NSRwere 0.87, 0.81 and 0.85, respectively, and the proportions of AF were0.12, 0.06 and 0.11, respectively. The most important differences lie inthe detected AF burdens—6% in the Tateno-Glass KS distance method and11% using the new COSEn measure. The burden of other rhythms is evenmore different—13% compared with 4%, respectively.

Another important finding is a sensible cut-off for detecting AF usingCOSEn. Both by visual inspection of the histogram and by analysis of theGaussian fit, we select COSEn=−1 as a threshold value, and we classifyrecords with lower values as NSR and higher values as AF. Analysis ofthe areas of the components of the sum of 3 Gaussians fit suggest that11% of data are misclassified using COSEn compared with 28% using CV and8% using KS distances.

Example 3 Relates to AF Classification in 24-Hour Holter RecordingsUsing COSEn

FIGS. 1, 3, and 4 show three of the 24-hour Holter monitor recordings,and RR intervals are classified as NSR (blue) or AF (green) using onlyCOSEn, which is shown as a purple line at the bottom. The cut-off ofCOSEn=−1 was chosen by eye from inspection of the frequency histogram ofCOSEn values in 420 consecutive Holter recordings. FIGS. 3 and 4 showuninterrupted NSR and AF, respectively. FIG. 1 shows a record withparoxysms of AF. There is good agreement between the COSEn value and theappearance of the time series. Note that the y-axis is RR interval, andAF is characterized by faster rates (shorter intervals) as well asincreased variability.

Example 4—Finding Optimal Segmentation for MIT-BIH Arrhythmia Data Base

The minimum piecewise entropy algorithm was applied to the MIT-BIHArrhythmia Database prior to applying our logistic regression model topredict the presence of atrial fibrillation train on the MIT-BIH atrialfibrillation database. The algorithm was applied to pick the optimalchange points for segments with homogeneous mean and variance. Theoptimal number of change points was selected using a penalty based onthe number of segments and the length of the data record as previouslydescribed (See Lake D E. Efficient adaptive signal and signal dimensionestimation using piecewise projection libraries Wavelet Applications V,H. H. Szu, Editor, Proc. SPIE Vol. 3391, p. 388-395,1998., and Lake D E.Adaptive signal estimation using projection libraries (Invited Paper)Wavelet Applications IV, H. H. Szu, Editor, Proc. SPIE-3078, p.p.602-609, 1997.).

An example is shown in FIG. 6. The optimal number of segments was foundto be 18 with lengths ranging from 22 to 341 beats.

This procedure was repeated for all 48 subjects in the databaseresulting in 525 homogeneous segments. The moments and entropy for eachsegment were calculated and evaluated for the presence of AF.

Table 3, summarizes the results with a threshold of 0.8 and compares theresults to the method of Tateno and Glass (as summarized in Table 3 ofK. Tateno and L. Glass, “Automatic detection of atrial fibrillationusing the coefficient of variation and density histograms of RR and ΔRRintervals,” Med Biol Eng Comput vol. 39, 664-671, 2001.)

TABLE 3 Method TP TN FN FP Sens. Spec. Tateno-Glass 10218 89973 13716176 88.2% 93.6% COSEn 11534 94383 667 2910 94.5% 97.0%

Thus the new method has superior performance characteristics in thiscanonical database.

Example 5—Optimal Segmentation in Near-Real Time

This example demonstrates an alternative embodiment of the optimalsegmentation process according to the present invention. An advantage ofthis embodiment is exact identification of start and stop times of AF.To preserve near-real time performance, it is implemented only when therhythm is perceived to change into or out of AF.

An evaluation of the MIT ARH and AF databases using American NationalStandards ANSI/AAMI EC38:1998 was conducted to evaluate COSEnperformance on AF detection in the MIT ARH and AF databases. The 50previous and the 50 subsequent beats were used to identify homogenoussegments for classification using COSEn. This requires a delay of 50beats, and the algorithm is considered near-real time.

The following order of operations was employed: (1) COSEn was calculatedon non-overlapping 16 beat segments; (2) beats were labeled in eachsegment as AF or non-AF based on a threshold value determined from theUVa Holter database; (3) when the label changed, the near-real timesegmentation analysis was implemented to determine whether there was astatistically significant change in the rhythm and, if so, the exactbeat at which the label should change; (4) in the ARH database, anysegments with more than 25% of beats labeled as ectopic were classifiedand labeled as non-AF. Results are displayed in Tables 4 and 5. Notethat the output is in standard sumstats format.

TABLE 4 MIT ARH database results Test Record TPs FN TPp FP ESe E + P DSeD + P Ref duration duration 100 0 0 0 0 — — — — 0.000 0.000 101 0 0 0 0— — — — 0.000 0.000 102 0 0 0 0 — — — — 0.000 0.000 103 0 0 0 0 — — — —0.000 0.000 104 0 0 0 0 — — — — 0.000 0.000 105 0 0 0 0 — — — — 0.0000.000 106 0 0 0 0 — — — — 0.000 0.000 107 0 0 0 0 — — — — 0.000 0.000108 0 0 0 0 — — — — 0.000 0.000 109 0 0 0 0 — — — — 0.000 0.000 111 0 00 0 — — — — 0.000 0.000 112 0 0 0 0 — — — — 0.000 0.000 113 0 0 0 0 — —— — 0.000 0.000 114 0 0 0 0 — — — — 0.000 0.000 115 0 0 0 0 — — — —0.000 0.000 116 0 0 0 0 — — — — 0.000 0.000 117 0 0 0 0 — — — — 0.0000.000 118 0 0 0 0 — — — — 0.000 0.000 119 0 0 0 0 — — — — 0.000 0.000121 0 0 0 0 — — — — 0.000 0.000 122 0 0 0 0 — — — — 0.000 0.000 123 0 00 0 — — — — 0.000 0.000 124 0 0 0 0 — — — — 0.000 0.000 200 0 0 0 0 — 0— 0 0.000 1:31.952 201 3 0 2 0 100 100 100 73 10:05.800 13:46.688 202 30 3 0 100 100 81 88  9:31.080  8:47.475 203 15 0 1 0 100 100 100 9222:58.497 24:51.286 205 0 0 0 0 — — — — 0.000 0.000 207 0 0 0 0 — — — —0.000 0.000 208 0 0 0 0 — — — — 0.000 0.000 209 0 0 0 0 — — — — 0.0000.000 210 6 0 1 0 100 100 100 97 29:12.513 30:05.555 212 0 0 0 0 — — — —0.000 0.000 213 0 0 0 0 — — — — 0.000 0.000 214 0 0 0 0 — — — — 0.0000.000 215 0 0 0 0 — — — — 0.000 0.000 217 1 0 1 0 100 100 100 14 0:49.688  5:56.738 219 7 0 4 0 100 100 100 96 23:21.730 24:15.705 220 00 0 0 — — — — 0.000 0.000 221 8 0 1 0 100 100 100 93 27:57.755 30:05.555222 2 0 2 0 100 100 100 27  5:13.694 19:13.733 223 0 0 0 0 — — — — 0.0000.000 228 0 0 0 3 — 0 — 0 0.000  8:49.361 230 0 0 0 0 — — — — 0.0000.000 231 0 0 0 0 — — — — 0.000 0.000 232 0 0 0 0 — — — — 0.000 0.000233 0 0 0 0 — — — — 0.000 0.000 234 0 0 0 0 — — — — 0.000 0.000 Sum 45 015 4 2:09:10.757 2:47:24.048 Gross 100 79 98 76 Average 100 80 98 58Summary of results from 48 records

TABLE 5 MIT AF database results (AF detection) Test Record TPs FN TPp FPESe E + P DSe D + P Ref duration duration 00735 1 0 1 0 100 100 85 954:24.068 3:57.740 03665 6 0 5 2 100 71 100 98 1:39:12.612 1:41:13.71204015 2 0 1 12 100 8 100 10 3:22.116 33:43.156 04043 67 4 66 5 94 93 8892 2:08:19.984 2:02:40.980 04048 3 0 3 2 100 60 99 13 4:43.104 35:26.51604126 5 0 5 3 100 63 100 87 22:18.568 25:37.816 04746 2 0 2 0 100 100100 100 5:25:16.396 5:24:43.176 04908 6 1 5 4 86 56 93 91 51:04.02452:36.796 04936 26 3 94 0 90 100 70 99 7:21:33.528 5:13:13.296 05091 1 01 0 100 100 100 78 0:42.552 0:54.728 05121 16 1 38 1 94 97 91 986:25:57.448 5:57:21.352 05261 2 0 2 7 100 22 100 54 6:21.796 11:50.14006426 22 1 19 1 96 95 100 98 9:44:30.812 9:51:35.768 06453 2 1 2 1 67 6750 69 5:19.072 3:51.488 06995 3 1 7 9 75 44 98 97 4:48:44.4524:51:32.332 07162 1 0 1 0 100 100 100 100 10:13:42.344 10:13:43.04007859 1 0 55 0 100 100 92 100 10:13:42.868 9:24:53.036 07879 1 0 6 0 100100 99 100 6:09:59.948 6:08:02.492 07910 3 0 3 0 100 100 98 1001:37:33.584 1:36:15.664 08215 2 0 2 0 100 100 100 100 8:14:33.9688:14:08.084 08219 38 0 25 8 100 76 95 81 2:12:05.500 2:35:34.072 08378 31 3 8 75 27 93 21 25:39.508 1:53:33.364 08405 1 0 1 1 100 50 100 1007:22:51.916 7:24:05.300 08434 3 0 2 0 100 100 99 92 23:43.736 25:35.94408455 2 0 2 0 100 100 100 100 7:04:31.024 7:04:38.284 Sum 219 13 351 6493:10:14.928 92:50:48.276 Gross 94 85 95 96 Average 95 77 94 83 Summaryof results from 25 records

Excellent performance in the MIT ARH and AF databases does notnecessarily translate into robust real-world results, because of variousproblems with the MIT databases.

The RR interval time series in the ARH database of the 2 patients withAF with the results of our analysis compared with the EKGinterpretation. EKG waveforms from some of the disputed areas are shownin FIGS.

FIG. 8 shows the complete 30 minute RR interval time series from Record202, a complex record with obvious rhythm changes.

FIG. 9 shows our labeling strategy—blue dots are RR intervals in whichwe agree with the electrocardiographer who labeled the database, reddots are intervals we labeled as AF but s/he did not, and green dots wedid not label as AF but s/he did. The open bars below show duration ofAF episodes as labeled by the electrocardiographer (in green), and asdetected by our numerical algorithm (in purple).

Several areas are identified by circled numbers, and the correspondingEKG strips are shown.

Strip 1, in FIG. 10, shows sinus rhythm that we agree on.

Strip 2, in FIG. 10, was detected by us as AF but is obviously not AF—itis sinus rhythm with very, very frequent atrial ectopy. This rhythm issometimes a harbinger of AF, but is not AF. It is not surprising that wedetect it as AF because of its irregularity.

Strip 3, in FIG. 10, was not detected by us as AF but was labeled as AFby the electrocardiographer. Inspection of the entire record shows thatthe rhythm is AF with varying degrees of organization, or atrialflutter-like qualities. For this epoch, the atrial activity was ratherorganized resulting in a more regular ventricular rhythm, hence ourmisdiagnosis. Clinically, the patient would be treated for AF.

Strip 4, in FIG. 11, is clearly AF, and we agree.

Strip 5, in FIG. 11, is essentially identical to strip 3. We calledneither AF. The electrocardiographer called the first one AF and thesecond one atrial flutter. This is inconsistent, and emphasizes theproblem with using these databases as the gold standard for arrhythmiadiagnostics.

FIG. 12 shows the entire 30 minute RR interval time series for Record203. There is no obvious change in rhythm, and we detect AF throughout.The electrocardiographer found the section in the middle to be atrialflutter and the rest to be AF. In FIG. 13, strips 6 and 7 show, to oureye, identical rhythm that we would characterize as somewhat organizedAF with either PVCs or aberrantly conducted impulses. We see no reasonto call them different rhythms.

The conclusion is that the labeling of rhythm in the ARH database isopen, in some places, for discussion. A detection algorithm thatcorrectly follows all of the ARH labeling is, in our opinion, overfit.

A first limitation for arrhythmias other than AF is recognized. Falsenegatives are expected in atrial flutter and AF with a more organizedatrial activation—these are more regular rhythms and resistant todiagnosis by changes in entropy. This problem is recognized in theepicmp analysis, which excludes atrial flutter episodes from evaluationof AF detection. False positives are expected in frequent ectopy, atrialor ventricular—these are irregular rhythms for the most part, and willhave higher entropy. A second limitation is recognized with regard toarbitrary use of 30 seconds as a minimum length of AF. This is untestedfrom a clinical point of view, and it is possible that even these shortepisodes increase stroke risk. This problem is shared by all AFdetection strategies.

Example 6—AF Detection Using COSEn in the UVa Holter Database

This example focuses on the UVa Holter database. More specifically, thisexample examines COSEn distributions in consecutive patients. More than600 consecutive 24-hour Holter recordings, including digitizedwaveforms, RR intervals and rhythm labels from the Philips system areavailable. Furthermore, this example inspects of the EKG waveforms toverify the rhythm labeling in records of patients over 45 years of agein whom atrial fibrillation (AF) was detected. In addition, this exampleinvolves inspection of the first 100 consecutive records.

First, regarding AF detection using COSEn in the UVa Holter database,FIG. 14 shows histograms of COSEn of more than 700,000 16-beat segmentsfrom 114 24-hour records for which the rhythm labels of normal sinusrhythm (NSR) or AF have been corrected. There is reasonable separation,and a cut-off value of −1 is suggested from inspection. Note that thereis a little hump in the tail to the left in the histogram of COSEn inAF, centered on COSEn values around −2, indicating a more regular set ofRR intervals than the rest of the group. These data have been identifiedand the corresponding EKG will be inspected more closely. Because ofrecord selection, AF is over-represented in these histograms. Overall,about 10% of the data set seems to be AF.

We have characterized the rhythms as AF or not, based on (1) COSEncalculation of 16-beat segments and (2) near-real time segmentation, andcompared the results to our labeling from EKG inspection. The results inepicmp and sumstats format for the 22 patients over age 45 (of more than70 total) that have been inspected with at least some detected AF arepresented in Table 6.

TABLE 6 Record TPs FN TPp FP ESe E + P DSe D + P Ref duration Testduration 11 4 0 13 0 100 100 98 100 23:59:37.480 23:32:58.885 27 2 0 230 100 100 99 100 23:59:09.230 23:38:47.165 30 0 0 0 0 — — — — 0.0000.000 38 4 0 9 0 100 100 99 100 23:56:11.465 23:45:47.130 62 1 0 8 0 100100 100 100 23:59:13.065 23:52:55.730 67 1 0 73 0 100 100 45 10022:46:29.785 10:09:50.740 68 52 3 50 4 95 93 99 98 13:37:41.11513:45:55.700 74 2 0 3 0 100 100 100 100 23:58:23.050 23:58:39.525 87 1 01 0 100 100 100 100 23:59:49.880 23:59:49.880 137 4 0 1 0 100 100 100100 23:59:40.085 23:59:49.680 141 2 0 52 0 100 100 98 98 23:35:58.68523:25:20.400 144 1 0 1 0 100 100 100 100 23:59:50.010 23:59:50.010 147 10 74 0 100 100 95 100 23:59:50.045 22:43:55.675 153 1 0 1 0 100 100 100100 23:59:50.185 23:59:50.185 154 3 0 11 1 100 92 94 97 3:55:35.2353:50:10.340 250 3 0 154 0 100 100 59 100 23:59:39.715 14:13:52.625 282 20 2 5 100 29 99 94 1:26:52.315 1:31:34.550 296 1 0 15 0 100 100 99 10023:59:50.105 23:47:48.675 387 1 0 5 0 100 100 100 100 23:59:50.10523:59:09.835 391 2 0 24 66 100 27 42 47 3:03:06.560 2:41:43.370 1088 160 7 0 100 100 100 100 23:58:46.520 23:54:19.710 1090 2 0 9 0 100 100 100100 23:59:47.405 23:57:09.450 Sum 106 3 536 76 428:15:12.040402:49:19.260 Gross 97 88 94 100 Average 100 92 92 97 Summary of resultsfrom 22 records

By inspection of consecutive records, circumstances can be identified inwhich false positive and negative findings of AF occur. False positives(high entropy not due to AF) occur in circumstances of very frequentPVCs or PACs; very high heart rate variability in the young; SAWenckebach; and/or multifocal atrial tachycardia. False negativereadings of AF (low entropy despite AF) occur in circumstances of atrialflutter or organized AF with regular ventricular response (recall thatepicmp excludes atrial flutter from its analysis); and/or very slowheart rates.

As an example of the potential capability of the algorithm, results fora Holter showing paroxysmal AF are presented. In this small data set,this was the only record with false negative findings using COSEn.

FIG. 15 and FIG. 16 show the RR interval time series as points, and theAF episode durations as open bars below. FIG. 15 is the entire 24 hours,and FIG. 16 is the second hour of the recording. In these figures, bluepoints are RR intervals for which COSEn agrees with our owninterpretation of the EKG, whether NSR or AF; red points are intervalsthat COSEn labeled as AF but were not AF by EKG inspection; green pointsare AF intervals by EKG inspection that COSEn did not label as AF; andopen bars show duration of AF episodes by EKG inspection in green, andas detected by COSEn in purple.

Examination of FIG. 16, by eye, demonstrates that the agreement is good.Thus the major findings are that COSEn detects AF, and the segmentationassigns onsets and offsets accurately.

Example 7—Histograms of COSEn in the MIT and UVa Databases

FIG. 17 shows histograms of COSEn calculated in 16-beat segments fromthe entire MIT AF (top panels) and ARH (middle panels) databases andfrom the more than 100 UVa Holter recordings that we have overread(bottom panels). The left-hand panels are all rhythms other than AF, andthe right-hand panels are AF alone. The findings are of higher COSEn inAF, with similar properties in the MIT and UVa databases. Note thelabeling of the y-axes—the UVa database has more than 100 times as muchAF as the MIT ARH database, and is growing.

The histogram of COSEn in AF at UVa, presented in FIG. 17, differs fromthe one showed in EXAMPLE 6. The “small tail” to which attention wasdrawn in EXAMPLE 6, arose from a single patient. The RR interval timeseries of that record showed long segments of very regular rhythm, butno atrial activity was seen in the 3 EKG leads from the Holterrecording. A 12-lead EKG from that day, though, showed unmistakableatrial activity that was either atrial flutter or atrial tachycardia butwas not AF. This record (like all the records with atrial flutter) wasremoved from the AF set, which now numbers 29.

These findings suggest that the UVa database should be an acceptabletest bed for arrhythmia detection algorithms, and a candidate as asurrogate for the MIT databases for FDA approval.

Table 7 presents epicmp and sumstats for 29 UVa Holter recordings.

TABLE 7 Record TPs FN TPp FP ESe E + P DSe D + P Ref duration Testduration 11 4 0 13 0 100 100 98 100 23:59:37.480 23:32:58.885 27 2 0 230 100 100 99 100 23:59:09.230 23:38:47.165 30 0 0 0 0 — — — — 0.0000.000 38 4 0 9 0 100 100 99 100 23:56:11.465 23:45:47.130 62 1 0 8 0 100100 100 100 23:59:13.065 23:52:55.730 67 1 0 73 0 100 100 45 10022:46:29.785 10:09:50.740 68 52 3 50 4 95 93 99 98 13:37:41.11513:45:55.700 74 2 0 3 0 100 100 100 100 23:58:23.050 23:58:39.525 87 1 01 0 100 100 100 100 23:59:49.880 23:59:49.880 137 4 0 1 0 100 100 100100 23:59:40.085 23:59:49.680 141 2 0 52 0 100 100 98 98 23:35:58.68523:25:20.400 144 1 0 1 0 100 100 100 100 23:59:50.010 23:59:50.010 147 10 74 0 100 100 95 100 23:59:50.045 22:43:55.675 153 1 0 1 0 100 100 100100 23:59:50.185 23:59:50.185 154 3 0 11 1 100 92 94 97 3:55:35.2353:50:10.340 177 31 0 16 1 100 94 100 99 23:37:03.840 23:47:52.565 190 10 1 0 100 100 100 90 0:52.075 0:57.835 194 4 0 3 0 100 100 100 10023:51:00.810 23:50:56.540 200 2 0 4 0 100 100 100 100 23:59:48.45523:58:19.810 205 6 0 8 0 100 100 99 100 23:59:16.540 23:44:41.935 245 100 4 0 100 100 100 100 23:59:12.045 23:58:36.975 252 136 0 10 0 100 100100 98 23:32:00.440 23:54:58.890 282 2 0 2 5 100 29 99 94 1:26:52.3151:31:34.550 291 2 0 1 0 100 100 100 100 23:59:45.040 23:59:50.195 296 10 15 0 100 100 99 100 23:59:50.105 23:47:48.675 384 4 0 3 63 100 5 10028 2:49:00.090 10:04:12.690 387 1 0 5 0 100 100 100 100 23:59:50.10523:59:09.835 1088 16 0 7 0 100 100 100 100 23:58:46.520 23:54:19.7101090 2 0 9 0 100 100 100 100 23:59:47.405 23:57:09.450 Sum 297 3 408 74571:00:25.100 563:14:10.700 Gross 99 85 97 99 Average 100 93 97 97Summary of results from 29 records

Note that in Table 7, about 95% of the data is AF. Note also that record384 has a large number of false positives. The EKG shows very frequentatrial ectopy and a very variable rhythm, but nonetheless is not AF.

The finding is that COSEn measurements of 16-beat segments andimplementing a near-real time segmentation algorithm has excellentperformance in detecting AF in patients who are evaluated for thatdiagnosis.

The following references are hereby incorporated by reference herein intheir entirety:

[1] K. Tateno and L. Glass, “Automatic detection of atrial fibrillationusing the coefficient of variation and density histograms of RR and ΔRRintervals,” Med Biol Eng Comput vol. 39, 664-671, 2001.

[2] S. Pincus, “Approximate entropy as a measure of system complexity,”Proc. Natl. Acad. Sci., vol. 88, pp. 2297-2301, 1991.

[3] J. Richman and J. Moorman, “Physiological time series analysis usingapproximate entropy and sample entropy,” Amer J Physiol, vol. 278, pp.H2039-H2049, 2000.

[4] D. Lake, J. Richman, M. Griffin, and J. Moorman, “Sample entropyanalysis of neonatal heart rate variability,” Amer J Physiol, vol. 283,pp. R789-R797, 2002.

[5] M. Costa, A. Goldberger, and C. Peng, “Multiscale entropy analysisof complex physiologic time series,” Phys. Rev. Lett., vol. 89, no. 6,p. 068102, 2002.

[6] J. Beirlant, E. J. Dudewicz, L. Gyorfi, and E. C. van der Meulen,“Nonparametric entropy estimation: An overview,” International Journalof Math. Stat. Sci., vol. 6, no. 1, pp. 17-39, 1997.

[7] O. Vasicek, “A test for normality based on sample entropy,” Journalof the Royal Statistical Society, Series B, vol. 38, no. 1, pp. 54-59,1976.

[8] A. Hyvarinen, J. Karhunen, and E. Oja, Independent ComponentAnalysis. John Wiley and Sons, 2001.

[9] M. Jones and R. Sibson, “What is projection pursuit?” Journal of theRoyal Statistical Society, Series A, vol. 150, pp. 1-36, 1987.

[10] B. Ripley, Pattern recognition and neural networks. CambridgeUniversity Press, 1996.

[11] L. Breiman, J. Friedman, R. Olshen, and C. Stone, Classificationand Regression Trees. Monterey, Calif.: Wandswork and Brooks-Cole, 1984.

[12] R. T. and C. N., Goodness-of-Fit Statistics for DiscreteMultivariate Data. New York: Springer-Verlag, 1988.

[13] C. E. Shannon, “A Mathematical Theory of Communication”, BellSystem Technical Journal, vol. 27, pp. 379-423 & 623-656, July &October, 1948

[14] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis.Cambridge: Cambridge University Press, 1997.

[15] P. Brockwell and A. Davis, Time Series: Theory and Methods.Springer, 1991.

[16] J. Richman, “Sample entropy statistics,” Ph.D. dissertation,University of Alabama Birmingham, 2004.

[17] B. L. S. P. Rao, Nonparametric Functional Estimation. London:Academic Press, Inc., 1983.

[18] D. Erdogmus, K. Hild, J. Principe, M. Lazaro, and I. Santamaria,“Adaptive blind deconvolution of linear channels using Renyi's entropywith Parzen window estimation,” IEEE Transactions on Signal Processing,vol. 52, no. 6, pp. 1489-1498, 2004.

[19] Lake D E, Renyi entropy measures of heart rate Gaussianity. IEEETransactions on Biomedical Engineering, Volume 53(1):21-27, 2006.

[20] Lake D E. Efficient adaptive signal and signal dimension estimationusing piecewise projection libraries Wavelet Applications V, H. H. Szu,Editor, Proc. SPIE Vol. 3391, p. 388-395, 1998.

[21] Lake D E. Adaptive signal estimation using projection libraries(Invited Paper) Wavelet Applications IV, H. H. Szu, Editor, Proc.SPIE-3078, p.p. 602-609, 1997.

The following patents, applications and Publications are herebyincorporated by reference herein in their entirety:

U.S. Patent Publication No. 2005/0004486A1 to Glass et al., entitled“Detection of Cardiac Arrhythmia Using Mathematical Representation ofStandard Deltarr Probability Density Histograms;”

U.S. Patent Publication No. 2005/0165320A1 to Glass et al., entitled“Method and System for Detection of Cardiac Arrhythmia;”

International Patent Publication No. WO 03/077755A1 to Tateno et al.,entitled “Detection of Cardiac Arrhythmia Using MathematicalRepresentation of Standard Orr Probability Density Histograms;”

International Patent Publication No. WO 02/24068A1 to Glass et al.,entitled “Method and System for Detection of Cardiac Arrhythmia;”

European Patent No. EP1485018 to Tateno et al., entitled “Detection ofCardiac Arrhythmia Using Mathematical Representation of Standard DRRProbability Density Histograms;” and

European Patent Publication No. EP1322223A1 to Glass et al., entitle“(ENG) Method and System for Detection of Cardiac Arrhythmia.”

The invention claimed is:
 1. A method for analyzing at least one cardiacrhythm having an RR interval series, the method comprising:characterizing RR intervals by at least one dynamic parameter, includingentropy; combining the at least one dynamic parameter with at least onedensity parameter, including mean, standard deviation, or coefficient ofvariation (CV); classifying the at least one cardiac rhythm based on thecombination of the at least one dynamic parameter and the at least onedensity parameter; and generating a diagnostic output based on theclassification, wherein the method employs multivariate statisticalmodels.
 2. The method of claim 1, wherein the at least one dynamicparameter is selected from the group consisting of differentialquadratic Renyi entropy rate measured using the SampEn algorithm,normalized sample entropy (SE), non-normalized sample entropy (Q), andthe coefficient of sample entropy (COSEn).
 3. The method of claim 2,wherein the at least one density parameter is selected from the groupconsisting of standard deviation and coefficient of variation (CV)measured using histogram summary statistics.
 4. The method of claim 2,wherein the characterization of RR-intervals by the at least one dynamicparameter takes into account an order of data points in the RR-intervalseries.
 5. The method of claim 2, wherein the at least one densityparameter is considered as a continuous value, not as a range.
 6. Themethod of claim 2, comprising: optimizing accuracy and discriminatingcapability of the entropy measures by applying a continuity correction.7. The method of claim 2, comprising: estimating the standard error ofsample entropy and COSEn.
 8. The method of claim 2, comprising:identifying at least one segment of the at least one cardiac rhythm,wherein an RR interval distribution of the at least one segment displaysstatistical homogeneity.
 9. The method of claim 8, wherein thestatistical homogeneity is measured in terms of the mean, and/or thestandard deviation of the RR interval distribution.
 10. An apparatuscomprising: a programmable computer controller, programmed to: analyzeat least one cardiac rhythm, having an RR interval series bycharacterizing RR intervals by at least one dynamic parameter, includingentropy; combine the at least one dynamic parameter with at least onedensity parameter, including mean, standard deviation, or coefficient ofvariation (CV); and classify the at least one cardiac rhythm based onthe combination of the at least one dynamic parameter and the at leastone density parameter.